Restriction of a character sheaf to conjugacy classes

Abstract

Let A be a character sheaf on a connected reductive group G over an algebraically closed field. Assuming that the characteristic is not bad we show that for certain conjugacy classes D in G, the restriction of A to D is a local system up to shift. We also give a parametrization of unipotent cuspidal character sheaves of G in terms of restriction to conjugacy classes. Without restriction on characteristic we define canonical bijections from the set of unipotent representations of the corresponding split group over a finite field to a set combinatorially defined in terms of the Weyl group.